Functions as Power Series

MAT 229, Fall 2021

Week 12

Supporting materials

If you wish to get a different perspective on the notes below, try either of the following textbook sections.

Stewart’s Calculus
	Section 11.9: Functions as power series

Boelkins/Austin/Schlicker’s Active Calculus
	Section 8.6: Power series

Review

Questions

Write out the first four terms of each sequence.

f(x)=(((Underoverscript[Σ, (n=2), ∞])(-1))^n)(((n(x+3))^n)/(4^n))		(Video)

g(x)=(Underoverscript[Σ, (n=0), ∞])(((n(2x-5))^n)/((n^2)+1))		(Video)

h(x)=(Underoverscript[Σ, (n=3), ∞])((((x-10))^n)/(n^n))			(Video)

Find the center of convergence and radius of convergence for the above power series.

Find the domains of the functions defined by the above power series.

Functions as power series

Functions related to geometric series.

Questions

Consider the power series (Underoverscript[Σ, (n=0), ∞])(x^n).

For which values of x does this series converge?

What kind of series is this?

For the values of x when this series converges, what does it converge to?

(Video)

Questions

Consider the function with power series representation from above (1/(1-x))=(Underoverscript[Σ, (n=0), ∞])(x^n).

Replace the x in this series with  -x. In this new series for which values of x does the new series converge? What does it converge to?

Sum[(-x)^n,{n,0,Infinity}]

Replace the x in the original series with  ±(x^2). What happens?

Sum[(-x^2)^n,{n,0,Infinity}]

Replace the x in the original series with  x-1. What happens?

Sum[(x-1)^n,{n,0,Infinity}]

Replace the x in the original series with  -(x-1). What happens?

Sum[(-(x-1))^n,{n,0,Infinity}]

(Video)

Questions

What is a power series representation for ((x^3)/(1-x))=(x^3)(1/(1-x))?

x^3Sum[x^n,{n,0,Infinity}]

What is a power series representation for ((x^2)/(1+2x))?

x^2Sum[(-2x)^n,{n,0,Infinity}]

Calculus of power series

Differentiation

If  f(x)=(Underoverscript[Σ, (k=0), ∞])(((Subscript[c, k])(x-a))^k) has radius of convergence R, then f'(x)=(Underoverscript[Σ, (k=1), ∞])(Subscript[c, k])((k(x-a))^(k-1)) and it has radius of convergence R. However, you may lose convergence at one or both endpoints of the domain.

D[Sum[x^n,{n,0,Infinity}],x]

Integration

If  f(x)=(Underoverscript[Σ, (k=0), ∞])(((Subscript[c, k])(x-a))^k) has radius of convergence R, then ∫f(x)d x=(Underoverscript[Σ, (k=0), ∞])((Subscript[c, k])/(k+1))(((x-a))^(k+1))+C and it has radius of convergence R. You may gain convergence at one or both endpoints of the domain.

Integrate[Sum[x^n,{n,0,Infinity}],x]

Questions

Let h(x)=(1/(1+x)).

What is a power series representation for this function?

Sum[(-x)^n,{n,0,Infinity}]

For which values of x does this power series converge?

Integrate h(x) and its power series.

Integrate[Sum[(-x)^n,{n,0,Infinity}],x]
Series[Log[1+x],{x,0,4}]

For which values of x does this new power series converge?

(Video)

Approximate ln(1.5) with error less than 0.0001.		(Video)

Integrate[(-x)^n,x]
(* We'll use a comparison to obtain an n, with a geometric series of r=1/2: *)
soln=Solve[(1/2)^(n)==.0001,n]
enn=Ceiling[n/.soln[[1]]]
approx=Sum[(((((-(-0.5)))^n)(- 0.5))/(1+n)),{n,0,enn}]
exact=Log[0.5]
error=Abs[exact-approx]
TableForm[{{enn},{approx},{exact},{error}},TableHeadings->{{"n","Sn","S","Abs. Error"},{"ln(0.5)"}}]

Find a power series representation for f(x)=x ln(1+x) centered at 0.	(Video)

Sum[x Integrate[(-x)^n,x],{n,0,Infinity}]

Questions

Let g(x)=(1/(1+(x^2))).

What is a power series representation for this function?

Sum[(-x^2)^n,{n,0,Infinity}]

For which values of x does this power series converge?

Integrate h(x) and its power series.

Sum[Integrate[(-x^2)^n,x],{n,0,Infinity}]
Sum[(-1)^n x^(2n+1)/(2n+1),{n,0,Infinity}]

For which values of x does this new power series converge?

(Video)

Approximate (tan^(-1))(0.5) with error less than 0.0001.	(Video)

Integrate[(-x^2)^n,x]
soln=Solve[(1/2)^(2(n+1)+1)/(2(n+1)+1)==.0001,n]
enn=Ceiling[n/.soln[[1]]]
approx=Sum[(((((-1))^n)(((0.5))^(2n+1)))/(1+2n)),{n,0,enn}]
exact=ArcTan[0.5]
error=Abs[exact-approx]
TableForm[{{enn},{approx},{exact},{error}},TableHeadings->{{"n","Sn","S","Abs. Error"},{"(tan^(-1))(0.5)"}}]

Find a power series representation for h(x)=(((tan^(-1))(x))/x) centered at 0.	(Video)

You can now integrate  (tan^(-1))(x) in two ways: by parts, or by term-by-term integration. Each has its advantages.	(Video)

Integrate[ArcTan[x],x]

Questions

Let k(x)=(Underoverscript[Σ, (n=0), ∞])((x^n)/(n!)).

What is k(0)?

k[x_]:=Integrate[Sum[x^n/Factorial[n],{n,0,Infinity}],x]
k[0]

What is the domain of this function?

Write out the first 5 terms of k(x), the first 5 terms of k'(x) and the first 5 terms of ∫k(x)d x. What well known function is this?

Table[x^n/Factorial[n],{n,0,4}]
Integrate[Sum[x^n/Factorial[n],{n,0,4}],x]

(Video)

Homework

IMath problems on the absolute convergence test.